Question

Define the simple events for the experiments in Exercises \(16-20 .\) The grade level of a high school student is recorded.

Short answer

Answer: The simple events for the experiment are E1, E2, E3, and E4, representing grades 9, 10, 11, and 12, respectively.

Step-by-step solution

List possible grade levels

To define the simple events, we start by listing all possible grade levels a high school student may be in. These are typically grades 9, 10, 11, and 12.

Define the simple events

Each grade level can be considered a simple event in this experiment. We can represent these simple events as follows: - Grade 9: E1 - Grade 10: E2 - Grade 11: E3 - Grade 12: E4 So, the simple events for the experiment of recording the grade level of a high school student are E1, E2, E3, and E4, which correspond to grades 9, 10, 11, and 12, respectively.

Key concepts

Probability Theory

Probability theory is a branch of mathematics focused on analyzing random events. The core idea is to quantify the likelihood of various outcomes. It is built on the foundation of sets and their properties, as well as functions that assign probabilities to these sets.

When discussing probability in a high school setting, we consider events that are clearly defined outcomes of an experiment or a process. For example, flipping a coin has two possible outcomes: heads or tails. A crucial part of probability theory is understanding the different types of events—simple and compound.

A simple event is an event that cannot be broken down into more basic events. It is a single outcome of an experiment. In our textbook exercise example, the grade level of a high school student is recorded, and each grade level represents a simple event. These are mutually exclusive; a student can't be in Grade 9 and Grade 10 at the same time.

High School Statistics

High school statistics incorporates probability theory as one of its components and is concerned with collecting, analyzing, interpreting, and displaying data. Understanding the basics of probability is essential for high school students because it allows them to make sense of data and determine the likelihood of events.

For instance, in high school sports, statistics may be used to calculate the probability of a team winning a game based on past performance. By recognizing each game’s outcome as a simple event, students can apply probability concepts to real-life scenarios. When students learn to define simple events and calculate their probabilities, they are building fundamental skills that will help them in more complex statistical analyses, like determining the distribution of grades across a grade level or the average grade in a particular subject.

Probability Outcomes

Probability outcomes are the possible results of a random event. Each outcome has a probability assigned to it, which represents the chance of that outcome occurring. The probabilities of all possible outcomes in a given experiment must sum up to 1.

In our problem, the recording of a high school student's grade level, there are four distinct outcomes: E1, E2, E3, and E4. High school students can apply this concept by recognizing each of these outcomes as having a probability that can be calculated based on available data, such as the distribution of students across grade levels. Understanding simple events and their associated probabilities lays the groundwork for students to tackle more complex probability distributions and apply these principles to data analysis in their everyday lives.